Question

I am trying to solve this problem, but I don't know which test to use:

For average risk funds only, test the proportion of funds with Sales Charges is not 50% use α = 0.10. Explain your conclusion.

Here is the contingency table with the data:

		Sales charge
		Yes	No	total
Risk	Low	6	8	14
	Avg	42	34	76
	High	24	23	47
	Total	72	65	137

Context of the problem:

Mutual funds are the most common way people invest for the future with more than $20 trillion in investment assets. They are so popular because they are a simple, convenient way to invest in stocks, bonds, money markets, commodities and other securities. Nearly half of all Canadian households invest in mutual funds. These professionally managed investment products use money from investors to create a diversified pool of securities to achieve a specific investment objective. Investors purchase shares, which represent ownership in the mutual fund. The price of the fund fluctuates based on the gain or loss of the securities that make up the fund.

Because each mutual fund has a specific investment objective, such as growth, capital preservation, or income, it is important to understand which funds are designed to meet the investor objectives. For example Asset Allocation Funds invest in a mix of stocks, bonds, alternatives and money market securities, whereas TK Funds emphasize on the Science and Technology stocks.

Answer 1

First we find proportions of funds with Sales Charge when Average risk as:

• Count in cell of Average Risk and Yes for Sales charge. It is '42'.
• Then, we find Total of Average Risk funds=(42+34)=76
• Sample Proportion=phat=42/76=0.552632

It is required to test:

• Null hypothesis-H0: p=p0=0.5 [50%] VS Alternative hypothesi- H1: p is not equal to 0.5.
• where, p=population proportion, p0=specified value of proprtion
• It is testing of single proportion. It is known that, mean=p and variance=p*q are parameters of Bernoulli distribution.
• Here, we use normal approximation, Z=(phat-p0)/sqrt(p0*q0/n) will follow standard normal distribution.

Computation in MS-EXCEL:

Risk	Sales Charge
Yes	No	Total
Low	6	8	14
Avg	42	34	76
High	24	23	47
Total	72	65	137	N
	0.552632	0.447368	1
phat	0.552632	42/76
1-phat	0.447368	1-42/76
p0	0.5	50%
q0=1-p0	0.5
p0*q0	0.25
H0: p=p0=0.5 VS H1: p≠p0
Zcalculated	0.917663	(phat-p0)/sqrt(p0*q0/n)	HERE, n=76
p-value	0.820602	NORMDIST(0.105263,0,1,2)
alpha	0.1
Zcritical	1.281552	NORMINV(1-0.1,0,1)
Since, Zcalculated < Zcritical, we Fail to Reject H0.

Conclusion: For average risk funds only, the proportion of funds with Sales Charges is 50% use α = 0.10.